Local Rigidity of Teichm\"uller space with Thurston metric

TL;DR

This paper proves that isometries of cotangent spaces in Teichmüller space with the Thurston metric are induced by surface isometries, extending Royden's theorem to this setting.

Contribution

It establishes a rigidity result for the Thurston metric on Teichmüller space, showing isometries correspond to surface isometries, a new analogue of Royden's theorem.

Findings

Every R-linear surjective isometry arises from a surface isometry

Extension of Royden's theorem to Thurston metric

Rigidity of the Thurston metric in Teichmüller space

Abstract

We show that every $\mathbb R$-linear surjective isometry between the cotangent spaces to the Teichm\"uller space equipped with the Thurston norm is induced by some isometry between the underlying hyperbolic surfaces, which is an analogue of Royden's theorem concerning the Teichm\"uller metric.